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Brauer F., Nohel J.A. — The qualitative theory of ordinary differential equations
Brauer F., Nohel J.A. — The qualitative theory of ordinary differential equations



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Название: The qualitative theory of ordinary differential equations

Авторы: Brauer F., Nohel J.A.

Аннотация:

Superb, self-contained graduate-level text covers standard theorems concerning linear systems, existence and uniqueness of solutions, and dependence on parameters. Major focus on stability theory and its applications to oscillation phenomena, self-excited oscillations and regulator problem of Lurie. Bibliography. Exercises.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1969

Количество страниц: 314

Добавлена в каталог: 29.05.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
A priori bound      37
Abel's formula      46 48 57
Abel's theorem      100
Absolute stability      267—273
Acceleration      1
Acceleration, due to gravity      2
Air resistance      3 5 8 9 201
Almost linear system      144 160—171 184 185 201 261
Almost linear system, perturbed      237
Almost linear system, stability of      143—151 160—171
Amplitude      238 239 244 245 251
Asymptotic behavior of solutions      159 165 178
Asymptotic behavior of solutions of linear systems with constant coefficients      80—83
Asymptotic equivalence      144 159 178—183
Asymptotic stability      147 150—152 155 159 160 169 170 180 184 197 199 202 205 209 232 234
Asymptotic stability, criteria of      185
Asymptotic stability, extent of      215—228
Asymptotic stability, global      215—228 271 272
Asymptotic stability, of equilibrium solution      146
Asymptotic stability, of unperturbed system      169
Asymptotic stability, of zero solution      150 151
Asymptotic stability, region of      149 168 187 196 201 215—220 222 223
Asymptotic stability, uniform      147 169
Attractor      95 103 154 163 164
Autonomous equation      149 188
Autonomous perturbations      183
Autonomous systems      83—95 145 149 150 159 185 192 193 195 209 211 234 256
Autonomous systems, nonlinear      160
Autonomous systems, perturbed      163
Autonomous systems, stability of      192
Autonomous systems, two-dimensional      163 171
Averaging, method of      237 261
Basis      62 63 76 274 275 280 281
Basis for vector space      42
Bessel equation      180
Bessel functions      180
Boundary      25
Boundary points      25
Boundedness      223 225
Canonical form      77 78 171 274—283 290 292
Canonical form, diagonal      76 275
Canonical form, Jordan      73—80 279 282 283
Canonical form, of $2\times2$ matrices      284—289
Canonical form, real      292
Cauchy convergence criterion      56 131
Center      95 103 164
Chain rule      110 123 194
Change of variable      73
Characteristic exponents      98—101 159
Characteristic function      262
Characteristic function, admissible      262 263 264 267 269 273
Characteristic function, Characteristic polynomial      60 68 71 74 152 155 278 279
Characteristic function, linear      271
Characteristic roots      See Eigenvalues
Characteristic values      60
Circuit      10
Coefficient matrix      67 70 75 95
Cofactors      35
Column vector      15 33 34 45
Comparison Test      116
components      34
Components of region      197 211
Components of set      211
Conditional stability      171—178
Conditional stability of zero solution      172
Conservation of energy      189
Conservative mechanical system      188 189
Constant coefficients      55
Continuation      130 131 133 192 206
Continuation of solutions      127—135
Continuity      24—30
Continuity of solutions      28 30 108
Continuity of vector functions      19
Continuous dependence on initial conditions      137 148
Continuous dependence on initial values      212
Control      261
Control system      267 272
Control theory      223
Control, admissible      270
Controllability      237
Convergence      19
Convergence, absolute      116
Convergence, of successive approximations      114
Convergence, uniform      116 134
Critical points      85 86 90 95 145 147 160 187 191 192 200 201 211 252 264 269
Critical points, isolated      191 197
Critical points, stability of      184
Critical points, unstable      253
Current      9
Damping      225 251
Damping term      191
Dependence on initial conditions and parameters      135—142
Derivative of V with respect to system      193 194 209
Derivative of vector function      20
Determinant      34 35 47—49 154 155 265 296
Determinism      23
Diagonal matrix      See Matrix
Differentiability, of vector functions      19
DIMENSION      65 274 277—279 281
Dimension of vector space      41
Direct control      272
Direct control problem      273
Direct method      187
Direct sum      277 279
Direct sum of subspaces      64
Direction field      259
Dirichlet      187
Dissipation of energy      191
Distance between two vectors      19
Distance function      19
Dynamical system      85
Eigenvalues      60—65 70 71 74—76 81 90 95 98 100 104 105 151—153 156 158 159 161—165 171 172 179—181 183 185 200 261 264 265 270 271 275—279 287 284 300
Eigenvalues, complex      287 288
Eigenvalues, distinct      76 78 276
Eigenvalues, double      77 286
Eigenvalues, generalized      276
Eigenvalues, multiple      77 81 153
Eigenvalues, multiplicity $n_j$      64
Eigenvalues, multiplicity of      65 66 68 76 80 278 282 290
Eigenvalues, multiplicity two      61
Eigenvalues, of matrix      60
Eigenvalues, simple      77
Eigenvector      61—63 75 76 274 275 278 279 287 288
Eigenvector, corresponding      61—63 276
Eigenvector, generalized      See Generalized eigenvectors
Eigenvector, linearly dependent      275 277 286
Eigenvector, linearly independent      62—64 75 76 78 275 276 286
Electrical system      205
Elliptic integral      245
Energy function      238 245 247 248 252 254
Equation of motion      5 8 9
Equation of n-th order      14
Equation of pendulum      6
Equilibrium      4 9 107 146 151 168 169 197
Equilibrium point      145 190 191 197 250
Equilibrium point, asymptotically stable      147
Equilibrium point, conditionally stable      178
Equilibrium point, Equilibrium position      3 8 146 196
Equilibrium point, stable      147
Equilibrium point, unstable      147 178 245
Equilibrium solution      145 198
Equilibrium solution, asymptotically stable      146
Equilibrium solution, stable      146
Equilibrium solution, unstable      146
Equilibrium state      86 145
Equivalent integral equation      156
Equivalent system      13 14 73 95 153 220 223 224 228 230 233 248 267
Equivalent system of first-order equations      54 105
Equivalent system of n first-order equations      54
Euclidean distance      19
Euclidean length      17 18
Euclidean norm      24 205
Euclidean space      10 16 26
Existence and uniqueness for linear systems      37—39
Existence and uniqueness of solutions      29
Existence of solutions      24—30 108 111 119 120 130 133 145
Existence of solutions, bounded      179
Existence of solutions, of initial value problem      118
Existence of solutions, of scalar differential equations      108—122
Existence of solutions, periodic      101 237
Existence problem      23
Existence theory      108—142
Existence theory for systems of first-order equations      122—124
Experimental error      164
Exponential of matrix      55 79
Feedback      272
Feedback control signal      262
Feedback electronic circuit      251
Finite basis      42
Fixed point theorem      237
Floquet's theorem      96 97 185
Forcing term      51
Forcing vector, periodic      102
Friction      251
Function V      193 195 201 202 209 211 213 217 221 223 224 227 228
Function V, negative definite      193 195 201 202 224 229 231 232 234 266 269
Function V, positive definite      193 195 196 198 200 202 205 207—209 211 214 215 217 221 224 228—232 234 268 269
Function, linear independence of      41
Function, vector-valued      20 21
Fundamental matrix      45 46 48—55 57 60 62 64 66 67 69—72 74 78 80 86 96 99 102—105 151 152 159
Fundamental set of solutions      44
Fundamental theorem of calculus      24 53 109 241
General solution      55 71 73 104 105 107
Generalized eigenvalues      276
Generalized eigenvectors      274—283
Generalized eigenvectors, index of      277
Global result      134
Gravitational attraction      3 5
Gravitational force      2
Gravity      8
Gronwall inequality      30—32 38 125 126 136 137 140 156 157 162 165
Hamiltonian form      189
Hamiltonian systems      169 189
Homogeneous equation      54
Homogeneous system      See Linear systems
Hooke's law      3 5 7 9
Identity matrix      34 55 77 102
Improper integral      240
Improper node      90 93
Index, theory of      237
Indirect control      272
Inductance      10
Infinitesimal upper bound      231 232 234
Inhomogeneous system      See Linear systems
Initial condition      4 9 21—23 26 30 37 38 43 53 54 108 110 124 134 135 162 176 192 205
Initial condition, continuity of solution with respect to      24
Initial displacement      247
Initial position      250
Initial value      9 196
Initial value problem      9 11 26—28 72 109 110 124 126 145 175 192
Initial velocity      3 247 250
Instability      152 196 205
Instability of equilibrium solution      146
Integrability of vector functions      19
Integral equation      109—111 116—118 124 130 131 137 141 162 173 174 176 179 182 183
Integral equation of Volterra type      109
Integral equation, equivalent to initial value problem      110
Integral of system      189
Invariance under translations of time      84 85 147 192 205
Invariant      76
Invariant set      208—216 234
Invariant subset      215 218 222 224 235 236
Invariant subspace      274—283
Inverse matrix      See Matrix
Jordan canonical form      73—80 279 282 283
Jordan canonical matrix      282 283
kinetic energy      188 194 197
Kirchoff's law      9 106
l'Hopital's rule      203
Lagrange      187
Lagrange stability      223 224
Lagrange's theorem      188 190
Laplace transform      82
Levinson, N.      180
Lienard      259
Lienard equation      191 201 202 205 209 214 217 219 224 225 237
Lienard equation, periodic solutions of      250—261
Limit cycle      222
Limit point of orbit      211
Limiting autonomous system      235 236
Linear approximations      187
linear combination      41
Linear combination of solutions      44
Linear combination of vectors      42
Linear dependence      41
Linear dependence of functions      41
Linear differential equations, first-order      12
Linear differential equations, second-order      13
Linear homogeneous scalar equation      45
Linear independence      41 44 64 274
Linear independence of functions      41
Linear systems      34 37 65 66 80 95 96 133 149 151—159 161 200
Linear systems, algebraic      277
Linear systems, control of      263
Linear systems, homogeneous      39—53 55 72 102 107
Linear systems, homogeneous, algebraic      63
Linear systems, homogeneous, associated      54
Linear systems, homogeneous, corresponding      51 54 55
Linear systems, homogeneous, of first-order equations      50
Linear systems, inhomogeneous      54 55
Linear systems, nonhomogeneous      51—55 101 104 107
Linear systems, nonhomogeneous, of algebraic equations      102
Linear systems, of differential equations      33
Linear systems, stability of      143—159
Linear systems, stability properties of      187
Linear systems, two-dimensional      147
Linear systems, two-dimensional, with constant coefficients      89
Linear systems, unperturbed      169 170
Linear systems, with constant coefficients      55—74 82 84 89 98 185
Linear systems, with periodic coefficients      50 96—107 185 290
Linear systems, with variable coefficients      155
Linear unperturbed equation      170
Linearization      144
Linearized system      200
Linearly dependent eigenvectors      275 277 286
Linearly dependent set of vectors      41
Linearly independent columns      48
Linearly independent eigenvectors      62—64 75 76 78 275 276 286
Linearly independent points      43
Linearly independent set of vectors      41—44
Linearly independent solutions      43—45 54 62 64 69 70 153 172
Linearly independent solutions of homogeneous equations      54
Liouville transformation      141
Lipschitz condition      112 115 117 119 123 125 126 139 140
Lipschitz constant      113 123 140
Local existence theorem      114
Local problem      108
Logarithm of matrix      50 290—295
Lur'e problem      237
Lyapunov second method      150 169 187—236 262 266
Lyapunov theorem      191—208 231
Lyapunov theorem, on matrices      265 266 299
Lyapunov theorem, proof of      205—208
Lyapunov, A.M.      143 187
Mass      7 14
Mass-spring system      1—7 9 22 201
Mass-spring system, coupled      7—10 30
Mathematical model      1 3 6 23 24 108
Mathematical model for mass-spring system      5 22
Matrix      15 33 37
1 2
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